自然哲学的数学原理-61

an hundred fold and yet no tails are seen; that the light of the planets isyet more copious without any tail, but that comets are seen sometimeswith huge tails when the light of their heads is but faint and dull;forso it happened in the comet of the year 1680, when in the month of DeTHE SYSTEM OF THE WORLD. 557cember it was scarcely equal in light to the stars of the second magnitudeand yet emitted a notable tail, extending to the length of 40, 50, 60. or70, and upwards ; and afterwards, on the 27th and 28th of January, thehead appeared but as a star of the seventh magnitude ; but the tail (aswas said above), with a light that was sensible enough, though faint, wasstretched out to 6 or 7 degrees in length, and with a languishing lightthat was more difficultly seen, even to 12 and upwards. But on the 9thand 10th of February, when to the naked eye the head appeared no more,I saw through a telescope the tail of 2 in length. But farther : if thetail was owing to the refraction of the celestial matter, and did deviatefrom the opposition of the sun, according as the figure of the heavens requires, that deviation, in the same places of the heavens, should be alwaysdirected towards the same parts : but the comet of the year 1680, December 28 1. SJ1. P. M. at London, was seen in Pisces, 8 41 , with latitudenorth 28 6 , while the sun was in Capricorn 18 26 . And the comet ofthe year 1577, December 29, was in Pisces 8 41 , with latitude north2SD 40 ; and the sun, as before, in about Capricorn 18 26 . In bothcases the situation of the earth was the same, and the comet appeared inthe same place of the heavens ; yet in the former case the tail of the comet(as well by my observations as by the observations of others) deviatedfrom the opposition of the sun towards the north by an angle of 4| degrees, whereas in the latter there was (according to the observation ofTycht] a deviation of 21 degrees towards the south. The refraction,therefore, of the heavens being thus disproved, it remains that the phaenomeriaof the tails of comets must be derived from some reflecting matter.That vapours sufficient to fill such immense spaces may arise from thecomet s atmospheres, may be easily understood from what follows.It is well known that the air near the surface of our earth possesses aspace about 1200 times greater than water of the same weight ; and therefore a cylindric column of air 1200 feet high is of equal weight with acylinder of water of the same breadth, and but one foot high. But acylinder of air reaching to the top of the atmosphere is of equal weightwith a cylinder of water about 33 feet high ; and therefore if from thewhole cylinder of air the lower part of 1200 feet high is taken away, theremaining upper part will be of equal weight with a cylinder of water 32feet high. Wherefore at the height of 1200 feet, or two furlongs, theweight of the incumbent air is less, and consequently the rarity of thecompressed air greater, than near the surface of the earth in the ratio of33 to 32. And, having this ratio, we may compute the rarity of the airin all places whatsoever (by the help of Cor. Prop. XXII, Book II), supposing the expansion thereof to be reciprocally proportional to its compression ; and this proportion has been proved by the experiments of Hookeand others. The result of the computation I have set down in the follow558 THE SYSTEM CF THE WORLD.ing table, in the first column of which you have the height o the air inmiles, whereof 4000 m:ike a semi-diameter of the earth; in the second thecompression of the air, or the incumbent weight ; in the third its rarity orexpansion, supposing gravity to decrease in the duplicate ratio of thedistances from the earth s centre. And the Latin numeral charactersare here used for certain numbers of ciphers, as 0,xvii 1224 forIMJ00000000000000001224, and 26950 xv for 26956000000000000000,AlR sBut from this table it appears that the air, in proceeding upwards, israrefied in such manner, that a sphere of that air which is nearest to theearth, of but one inch in diameter, if dilated with that rarefaction whichit would have at the height of one semi-diameter of the earth, would fill allthe planetary regions as far as the sphere of Saturn, and a great way beyond ; and at the height of ten semi-diameters of the earth would fill upmore space than is contained in the whole heavens on this side the fixedstars, according to the preceding computation of their distance. Andthough, by reason of the far greater thickness of the atmospheres of comets,and the great quantity of the circum-solar centripetal force, it may happenthat the air in the celestial spaces, and in the tails of comets, is not sovastly rarefied, yet from this computation it ^s plain that a very smallquantity of air and vapour is abundantly sufficient to produce all the appearances of the tails of comets; for that they are indeed of a very notablerarity appears from the shining of the stars through them. The atmosphere of the earth, illuminated by the sun s light, though but of a few milesin thickness, obscures arid extinguishes the light not only of all the stars,but even of the moon itself; whereas the smallest stars are seen to shinethrough the immense thickness of the tails of comets, likewise illuminatedby the sun, without the least diminution of their splendor.Kepler ascribes the ascent of the tails of comets to the atmospheres oftheir heads, and their direction towards the parts opposite to the sun to theaction of the rays of light carrying along with them the matter of thecomets tails; and without any great incongruity we may suppose that, inso free spaces, so fine a matter as that of the aether may yield to the actionTHE SYSTEM OF THE WORLD. 559of the rays of the sun s light, though those rays are not able sensibly to movethe gross substances in our parts, which are clogged with so palpable a resistance. Another author thinks that there may be a sort of particles ofmatter endowed with a principle of levity as well as others are with apower of gravity ; that the matter of the tails of comets may be of theformer sort, and that its ascent from the sun may be owing to its levity ;but, considering the gravity of terrestrial bodies is as the matter of thebodies, and therefore can be neither more nor less in the same quantity ofmatter, I am inclined to believe that this ascent may rather proceed fromthe rarefaction of the matter of the comets tails. The ascent of smoke ina chimney is owing to the impulse of the air with which it is entangled.The air rarefied by heat ascends, because its specific gravity is diminished,and in its ascent carries along with it the smoke with which it is engaged./Vnd why may not the tail of a comet rise from the sun after the samemanner? for the sun s rays do not act any way upon the mediums whichthey pervade but by reflection and refraction ; and those reflecting particles heated by this action, heat the matter of the aether which is involvedwith them. That matter is rarefied by the heat which it acquires, andbecause by this rarefaction the specific gravity, with which it tendedtowards the sun before, is diminished, it will ascend therefrom like a stream,and carry along with it the reflecting particles of which the tail of thecomet is composed ; the impulse of the sun s light, as we have said, promoting the ascent.But that the tails of comets do arise from their heads (p. 488), and tendtowards the parts opposite to the sun, is farther confirmed from the lawswhich the tails observe ; for, lying in the planes of the comets orbits whichpass through the sun, they constantly deviate from the opposition of thesun towards the parts which the comets heads in their progress along thoseorbits have left; and to a spectator placed in those planes they appear inthe parts directly opposite to the sun ;but as the spectator recedes fromthose planes, their deviation begins to appear, and daily becomes greater.And the deviation, c&teris paribits, appears less when the tail is more oblique to the orbit of the comet, as well as when the head of the comet approaches nearer to the sun .; especially if the angle of deviation is estimatednear the head of the comet. Farther; the tails which have no deviationappear straight, but the tails which deviate are likewise bended into a certain curvature ; and this curvature is greater when the deviation is greater,and is more sensible when the tail, cccteris paribus, is longer; for in theshorter tails the curvature is hardly to be perceived. And the angle ofdeviation is less near the comet s head, but greater towards the other endof the tail, and that because the lower side of the tail regards the partsfrom which the deviation is made, and which lie in a right line drawn outinfinitely from the sun through the comet s head. And the tails that are560 THE SYSTEM OF THE WORLD.longer and broader; and shine with a stronger light, appear more resplendentand more exactly defined on the convex than on the concave side. Uponwhich accounts it is plain that the phenomena of the tails of comet? depend upon the motions of their heads, and by no means upon the places ofthe heavens in which their heads are seen ; and that, therefore, the tailg ofthe comets do not proceed from the refraction of the heavens, but fromtheir own heads, which furnish the matter that forms the tail;for as inour air the smoke of a heated body ascends either perpendicularly, if thebody is at rest, or obliquely if the body is moved obliquely, so in theheavens, where all the bodies gravitate towards the sun, smoke and vapourmust (as we have already said) ascend from the sun, and either rise perpendicularly, if the smoking body is at rest, or obliquely, if the body, in theprogress of its motion, is always leaving those places from which the upperor higher parts of the vapours had risen before. And that obliquity willbe less where the vapour ascends with more velocity, to wit, near thesmoking body, when that is near the sun ;for there the force of the sun bywhich the vapour ascends is stronger. But because the obliquity is varied,the column of vapour will be incurvated ; and because the vapour in thepreceding side is something more recent, that is, has ascended somethingmore lately from the body, it will therefore be something more dense onthat side, and must on that account reflect more light, as well as be betterdefined ;the vapour on the other side languishing by degrees, and vanishing out of sight.But it is none of our present business to explain the causes of the appearances of nature. Let those things which we have last said be true orfalse, we have at least made out, in the preceding discourse, that the raysof light are directly propagated from the tails of comets in right linesthrough the heavens, in which those tails appear to the spectators whereverplaced ; and consequently the tails must ascend from the heads of the cometstowards the parts opposite to the sun. And from this principle we maydetermine anew the limits of their dis-<~ tances in manner following. Let S representthe sun, T the earth, STA theelongation of a comet from the sun, andATB the apparent length of its tail;and because the light is propagated fromthe extremity of the tail in the directionof the right, line TB, that extremitymust lie somewhere in the line TB.Suppose it in D, and join DS cuttingTA in C. Then, because the tail is al -ways stretched out towards the partsnearly opposite to the sun, and there! oreTHE SYSTEM OF THE WORLD. 561the sun, the head of the comet, and the extremity of the tail, lie in a rightline, the comet s head will be found in C. Parallel to TB draw SA, meeting the line TA in A, arid the comet s head C must necessarily be foundbetween T and A, because the extremity of the tail lies somewhere in theinfinite line TB ; and all the lines SI) which can possibly be drawn fromthe point S to the line TB must cut the line TA somewhere between Tand A. Wherefore the distance of the comet from the earth cannot exceedthe interval TA. nor its distance from the sun the interval SA beyond, orST on this side the sun. For instance : the elongation of the comet of16SO from the sun, Dec. 12, was 9, and the length of its tail 35 at least.If, therefore, a triangle TSA is made, whose angle T is equal to the elongation 9, and angle A equal to ATB, or to the length of the tail, viz., 35,then SA will be to ST, that is, the limit of the greatest possible distanceof the comet from the sun to the semi -diameter of the oj-bis magnus, asthe sine of the angle T to the sine of the angle A, that is, as about 3 to11. And therefore the comet at that time was less distant from the sunthan by T3T of the earth s distance from the sun, and consequently eitherwas within the orb of Mercury, or between that orb and the earth. Again,Dec. 21, the elongation of the comet from the sun was 32f , and the lengthof its tail 70. Wherefore as the sine of 3^| to the sine of 70, that is,as 4 to 7, so was the limit of the comet s distance from the sun to the distance of the earth from the sun, and consequently the comet had not thengot without the orb of Venus. Dec. 28, the elongation of the comet fromthe sun was 55, and the length of its tail 56 ; and therefore the limit ofthe comet s distance from the sun was not yet equal to the distance of theearth from the same, and consequently the comet had not then got withoutthe earth s orbit. But from its parallax we find that its egress from theorbit happened about Jan. 5, as well as that it had descended far withinthe orbit of Mercury. Let us suppose it to have been in its perihelionDec. the 8th, when it was in conjunction with the sun ; and it will followthat in the journey from its perihelion to its exit out of the earth s orbitit had spent 28 days ; and consequently that in the 26 or 27 days following, in which it ceased to be farther seen by the naked eye, it hadscarcely doubled its distance from the sun ; and by limiting the distancesof other comets by the like arguments, we come at last to this conclusion, that all comets, during the time in which they are visible by us,are within the compass of a spherical space described about the sun as acentre, with a radius double, or at most triple, of the distance of the earthfrom the sun.And hence it follows that the comets, during the whole time of theirappearance unto us, being within the sphere of activity of the circumsolar force, and therefore agitated by the impulse of that force, will (byCor. 1, Prop. XII, Book I, for the same reason as the planets) be made tc36562 THE SYSTEM OF THE WORLD.move in conic sections that have one focus in the centre of the sun, andby radii drawn to the sun, to describe areas proportional to the times ;forthat force is propagated to an immense distance, and will govern themotions of bodies far beyond the orbit of Saturn.There are three hypotheses about comets (p. 466) ; for some will have itthat they are generated and perish as often as they appear and vanish ;others, that they come from the regions of the rixed stars, and are seen byus in their passage through the system of our planets ; and, lastly, others,that they are bodies perpetually revolving about the sun in very eccentricorbits. In the first case, the comets, according to their different vel cities,will move in conic sections of all sorts; in the second, they will describehyperbolas, and in either of the two will frequent indifferently all quarters of the heavens, as well those about the poles as those towards theecliptic ;in the third, their motions will be performed in ellipses very eccentric, and very nearly approaching to parabolas. But (if the law of theplanets is observed) their orbits will not much decline from the plane ofthe ecliptic; and, so far as I could hitherto observe, the third case obtains;for the comets do, indeed, chiefly frequent the zodiac, and scarcely everattain to a heliocentric latitude of 40. And that they move in orbitsvery nearly parabolical, I infer from their velocity ;for the velocity withwhich a parabola is described is every where to the velocity with which acomet or planet may be revolved about the sun in a circle at the same distance in the subduplicate ratio of 2 to 1 (by Gor. VII, Prop. XVI) ; and,by my computation, the velocity of comets is found to be much aboutthe same. I examined the thing by inferring nearly the velocities fromthe distances, and the distances both from the parallaxes and the phaenornenaof the tails, and never found the errors of excess or defect in the velocities greater than what might have arose from the errors in the distances collected after that manner. But I likewise made use of the reasoning that follows.Supposing the radius of the nrbis magiius to be divided into 1000parts: let the numbers in the first column of the following table representthe distance of the vertex of the parabola from the sun s centre, expressedby those parts : and a comet in the times expressed in col. 2, will passfrom its perihelion to the surface of the spheie which is described aboutthe sun as a centre with the radius of the orbis magnus ; and in thetimes expressed in col. 3, 4, and 5, it will double, triple, and quadruple,that its distance from f.l:o sun.THE SYSTEM 0* THE WORLD. 563TABLE L[This table, here corrected, is made on the supposition that the earth sdiurnal motion is just 59 , and the measure of one minute loosely 0,2909,in respect of the radius 1000. If those measures are taken true, thetrue numbers of the table will all come out less. But the difference,even when greatest, and to the quadruple of the earth s distance fromthe sun, amounts only to 16h. 55 .]The time of a comet s ingress into the sphere of the orbis magnus, orof its egress from the same, may be inferred nearly from its parallax, bn1with more expedition by the followingTABLE II.564 THE SYSTEM OF THE WORLD.The ingress 01 a comet into the sphere of the orbis magnus, or itsegress from the same, happens at the time of its elongation from the sun,expressed in col. 1, against its diurnal motion. So in the comet of 1681.Jan. 4, O.S. the apparent diurnal motion in its orbit was about 3 5 , andthe corresponding elongation 71 J ; and the comet had acquired this elongation from the sun Jan. 4, about six in Ae evening. Again, in the year1680, Nov. 11, the diurnal motion of the comet that then appeared wasabout 4| ; and the corresponding elongation 79f happened Now. 10, alittle before midnight. Now at the times named these comets had arrivedat an equal distance from the sun with the earth, and the earth was thenalmost in its perihelion. But the first table is fitted to the earth s meandistance from the sun assumed of 1000 parts ; and this distance is greaterby such an excess of space as the earth might describe by its annual motionin one day s time, or the comet by its motion in 16 hours. To reduce thecomet to this mean distance of 1000 parts, we add those 16 hours to theformer time, and subduct them from the latter; and thus the former becomes Jan. 4d. 10 1. afternoon ;the latter Nov. 10, about six in the morning. But from the tenor and progress of the diurnal motions it appearsthat both comets were in conjunction with the sun between Dec. 7 and Dec.8 ; and from thence to Jan. 4d. 10h. afternoon on one side, and to Nov.10 . 6h. of the morning on the other, there are about 28 days. And somany days (by Table 1) the motions in parabolic trajectories do require.But though we have hitherto considered those comets as two, yet, fromthe coincidence of their perihelions and agreement of their velocities, it isprobable that in effect they were but one and the same ; and if so, theorbit of this comet must have either been a parabola, or at least a conicsection very little differing from a parabola, and at its vertex almost incontact with the surface of the sun. For (by Tab. 2) the distance of thecomet from the earth, Nov. 10, was about 360 parts, and Jan. 4, about630. From which distances, together with its longitudes and latitudes,we infer the distance of the places in which the comet was at those timesto have been about 280 : the half of which, viz., 140, is an ordinate to thecomet s orbit, cutting off a portion of its axis nearly equal to the radiusof the orbis magnus, that is, to 1000 parts. And, therefore, dividing thesquare of the ordinate 140 by 1000, the segment of the axis, we find thelatu$ rectum 19, 16, or in a round number 20 ; the fourth part whereof,5, is the distance of the vertex of the orbit from the sun s centre. But thetime corresponding to the distance of 5 parts in Tab. 1 is 27d. 16h. 7 . Ir.which time, if the comet moved in a parabolic orbit, it would have beencarried from its perihelion to the surface of the sphere of the orbis mag*nus described with the radius 1000, and would have spent the double ofthat time, viz., 55d. 8|h. in the whole course of its motion within thatsphere : and so in fact it did ; for from Nov. 10d. 6h. of the morning, thfTHE SYSTEM OF THE WORLD. OOCtime of the comet s ingress into the sphere of the orbis magnns, to Jan..4 1. 10h. afternoon, the time of its egress from the same, there are 55(1. 16h.The small difference of 7 u. in this rude way of computing is to be neglected, and perhaps may arise from the comet s motion being some smallmatter slower, as it must have been if the true orbit in which it was carried was an ellipsis. The middle time between its ingress and egress wasDecember Sd. 21. of the morning ; and therefore at this time the cometought to have been in its perihelion. And accordingly that very day, justbefore sunrising, Dr. Halley (as we said) saw the tail short and broad, butverybright, rising perpendicularly from the horizon. From the positionof the tail it is certain that the comet had then crossed over the ecliptic,and got into north latitude, and therefore had passed by its perihelion,which lay on the other side of the ecliptic, though it had not yet come intoconjunction with the sun ; and the comet [see more of this famous comet,p. 475 to 486] being at this time between its perihelion and its conjunction with the sun, must have been in its perihelion a few hours before;for in so near a distance from the sun it must have been carried with greatvelocity, and have apparently described almost half a degree every hour.By like computations I find that the comet of 1618 entered the sphereof the orbis maxims December 7, towards sun-setting ; but its conjunction with the sun was Nov. 9, or 10, about 28 days intervening, as in thepreceding comet ;for from the size of the tail of this, in wtrch it wasequal to the preceding, it is probable that this comet likewise did comealmost into a contact with the sun. Four comets were seen that year ofwhich this was the last. The second, which made its first appearanceOctober 31, in the neighbourhood of the rising sun, and was soon after hidunder the sun s rays, 1 suspect to have been the same with the fourth,which emerged out of the sun s rays about Nov. 9. To these we may addthe comet of 1607, which entered the sphere of the orbis mi^-tnis Sept.14, O.S. and arrived at its perihelion distance from the sun about October19, 35 days intervening. Its perihelion distance subtended an apparentangle at the earth of about 23 degrees, and was therefore of 390 parts.And to this number of parts about 34 days correspond in Tab. 1 . Farther ; the comet of 1665 entered the sphere of the orbis nta^tnts aboutMarch 17, and came to its perihelion about April 16, 30 days intervening.Its perihelion distance subtended an angle at the earth of about sevendegrees, and therefore was of 122 parts : and corresponding to this numberof parts, in Tab. 1, we find 30 days. Again ; the comet of 1 682 enteredthe sphere of the orbis magnus about Aug. 11, and arrived at its perihelion about Sep. 16, being then distant from the sun by about 350 parts, to